The screenshot shows the trigger for built-in SAP connector, I'm not sure what it shows to be missing. Maybe because we adjusted the name of the trigger after the preview?
The GA'ed SAP built-in connector indeed functions for trigger by SAP, and is available to both Azure-deployed apps and local Visual Studio Code testing. In the preview SAP built-in connector, we had a limitation for the SAP trigger and it was then limited to Visual Studio Code. This limitation is now resolved.
There is no difference in the amount of data transferred between a pull or a push model to retrieve data from SAP. However, there may be application-specific and ops-team-specific preferences for pull vs push model. For example, the receiving system may want to pull increment of the large dataset at its optimal pace. Or the ops-team may want to monitor the outcome of send batches from an SAP dashboard. Either approach is possible with the Logic Apps built-in SAP connector based on what fits the use case best. A cloud solution architect may be able to advise on a specific model after analyzing the application, users' and operators' needs.
I am not familiar with Labware LIMS and its protocols. You did mention XML, which is supported by Logic Apps. Furthermore, the Logic App SAP built-in connector already converts the SAP IDoc, BAPI and RFC payloads to XML. And Logic App has native support for XSLT transform, as well as a brand new Data Mapper to adapt the source and destination XML schemas. A priori it should be a good fit.